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Riemann surfaces and integrable systems

Lecturer: 
Course Type: 
PhD Course
Academic Year: 
2016-2017
Period: 
Jan - May
Duration: 
90 h
Description: 

Content:

Part 1

1.1Short review of the classical theory of finite-dimensional integrable systems
1.2 Be-Hamiltonian Geometry and Lax pair
1.3 The Toda system
1.4 The Korteweg de Vries equation: direct and inverse scattering on the line with decreasing initial data
1.5 Long time asymptotic for the solution of the KdV equation with decreasing initial data and Deift-Zhou steepest descent method
1.6 The  Cauchy problem for the KdV equation with periodic initial data and action-angle variables.

Part  2 Riemann Surfaces

2.1  Definition,  examples, and topological properties
2.2  Holomorphic and meromorphic functions and differentials on a Riemann surface.
2.3 Jacobi variety and Abel theorem
2.4 Riemann Roch theorem and applications
2.5 Theta function and Riemann vanishing theorem
2.6 Baker-Akhiezer function and periodic solution of integrable nonlinear PDEs

Location: 
Tuesday and Thursday in room 136, Wednesday in room 133.

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